<p>Understanding the mechanism of accumulating evidence over time in deliberate decision-making is crucial for both humans and animals. While numerous models have been proposed over the past few decades to characterize the temporal weighting of evidence, the dynamical principle governing the neural circuits in decision making remain elusive. In this study, we proposed a solvable rank-1 neural circuit model to address this problem. We first derived an analytical expression for integration kernel, a key quantity describing how sensory evidence at different time points is weighted with respect to the final decision. Based on this expression, we illustrated that how the dynamics introduced in the auxiliary space—namely, a subspace orthogonal to the decision variable—modulates the flow fields of decision variable through a gain modulation mechanism, resulting in a variety of integration kernel types, including not only monotonic ones (recency and primacy) but also non-monotonic ones (convex and concave). Furthermore, we quantitatively validated that integration kernel shapes can be understood from dynamical landscapes and non-monotonic temporal weighting reflects topological transitions in the landscape. Additionally, we showed that training on networks with non-optimal weighting leads to convergence toward optimal weighting. Finally, we demonstrate that rank-1 connectivity induces symmetric competition to generate pitchfork bifurcation. In summary, we present a solvable neural circuit model that unifies diverse types of temporal weighting, providing an intriguing link between non-monotonic integration kernel structure and topological transitions of dynamical landscape.</p>